Long-time dynamics of an epidemic model with nonlocal di ff usion and free boundaries

: In this paper, we consider a reaction-di ff usion epidemic model with nonlocal di ff usion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. [1] by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number R 0 arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If R 0 ≤ 1, we prove that the epidemic vanishes eventually. On the other hand, if R 0 > 1, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni [2] to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal di ff usion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. [1]).


Introduction
To model the 1973 cholera epidemic in the European Mediterranean region, Capasso and Paveri-Fontana [3] proposed the following ODE system u ′ (t) = −au(t) + cv(t), v ′ (t) = −bv(t) + G(u(t)), t > 0, (1.1) where • u(t) and v(t) represent respectively the average population concentration of the infectious agents and the infective humans in the infected area at time t, • a, b, c are all positive constants such that 1/a represented the mean lifetime of the agents in the environment, 1/b the mean infectious period of the infective humans, c the multiplicative factor of the infectious agents due to the infective humans, and • the function G(u) is the infection rate of the human population, assuming that the total number of susceptible humans remain constant during the epidemic. The function G is assumed to satisfy the following: (G1) G ∈ C 1 ([0, ∞]), G(0) = 0, G ′ (z) > 0 for all z ≥ 0; (G2) G(z) z ′ < 0 for z > 0 and lim z→+∞ G(z) z < ab c . A simple example of such a function is given by G(z) = αz 1+z with α ∈ (0, ab/c).
They were able to establish the following result for the long-time dynamics of (1.1): Let R 0 := cG ′ (0) ab ; then regardless of the positive initial values u(0) and v(0), (i) the epidemic tends to extinction if R 0 < 1, namely lim t→∞ (u(t), v(t)) → (0, 0) if R 0 < 1, (ii) the epidemic tends to a positive equilibrium state if R 0 > 1, namely lim t→∞ (u(t), v(t)) → (u * , v * ) if R 0 > 1, where u * , v * are uniquely determined by To include the mobility of the infectious agents (assuming the mobility of the infective human population is small and thus ignored), Capasso and Maddalena [4] proposed the following spatial reaction-diffusion model with Robin (or Neumann) boundary conditions u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω, (1. 3) where d denotes the diffusion rate of u, the epidemic region Ω ⊂ R N is a smooth bounded domain and α ≥ 0. They proved that the long-time behaviour of (1.3) is similar to the ODE model (1.1) with R 0 there replaced byR 0 := cG ′ (0) (a+dλ 1 )b , where λ 1 is the first eigenvalue of the eigenvalue problem in Ω, ∂ϕ ∂n + αϕ = 0 on ∂Ω.
In the literature, R 0 andR 0 are often called the reproduction number of the epidemic being modelled.
To describe the spatial spreading of an epidemic, it is important to know how the front of the epidemic propagates. In [5], Ahn, Baek, and Lin regarded the epidemic region as a changing interval and used the following free boundary problem to model the evolution and spreading of the epidemic: , h(t)), v t = −bv + G(u), t > 0, x ∈ (g(t), h(t)), (1. 4) where h(t) and g(t) are the moving boundaries of the infected region, µ is a positive constant and the initial data (u 0 , v 0 ) satisfy u 0 , v 0 ∈ C([−h 0 , h 0 ]), u 0 (±h 0 ) = v 0 (±h 0 ) = 0, and u 0 , v 0 > 0 in (−h 0 , h 0 ). (1.5) The equations for h ′ (t) and g ′ (t) mean that the expanding rate of the infected region is proportional to the spatial gradient of u at the front. This is known as the Stefan condition which was first used to describe the melting of ice (see, e.g., [6]). It has been extensively used in the study of the spread of population since Du and Lin [7]. The long-time dynamics of (1.4) can be described by a spreading-vanishing dichotomy; more precisely, Ahn et al. showed that the unique solution (u, v, g, h) to (1.4) satisfies either , h(t)) = R, and lim t→∞ (u(x, t), v(x, t)) = (u * , v * ) locally uniformly for x ∈ R.
Furthermore, using the reproduction number of (1.1), namely the dichotomy is determined as follows: If R 0 ≤ 1, then vanishing always happens; in the case where R 0 > 1, there exists a critical length, l * := π 2 d 1 a(R 0 −1) , such that • if h 0 ≥ l * (i.e., the initial size of the infected region is no less than 2l * ), then spreading always happens, and • if h 0 < l * , then vanishing (resp. spreading) happens if the initial functions (u 0 , v 0 ) are sufficiently small (resp. large).
In the case of an epidemic spreading predicted by (1.4), it was shown by Zhao, Li, and Ni [8] that there exists a uniquely determined c 0 > 0 such that which means the epidemic region [g(t), h(t)] expands with asymptotic speed c 0 . In (1.4) (as well as in (1.3)), the spatial dispersal of the infectious agents is assumed to follow the rules of random walk, which ignores any nonlocal effect in the dispersal process. Such nonlocal effect can be included if the local diffusion operator is replaced by a nonlocal diffusion operator of the form with an appropriate kernel function J. Here J(x) can be interpreted as the probability that an individual of the species moves from location 0 to x. A widely used class of kernel functions consists of J : R → R satisfying One recent paper by Cao, Du, Li, and Li [9] extended many basic results of [7] to the corresponding nonlocal model with the above kernel. Following the fashion of [9], Zhao, Zhang, Li, and Du [1] considered the corresponding nonlocal version of (1.4), which has the following form It was shown in [1] that (1.7) has a unique solution defined for all t > 0, and its long-time dynamics is determined by a spreading-vanishing dichotomy, in a similar fashion to (1.4) (with some subtle differences though). A striking difference of (1.7) to (1.4) is revealed by [2], which shows that the spreading determined by (1.7) may have infinite asymptotic spreading speed, a phenomenon known as "accelerated spreading". More precisely, if the kernel function J satisfies and spreading happens, then Moreover, if J(x) ≃ |x| −γ for some γ ∈ (1, 2] and all large |x| > 0, then for all large t > 0, Here, and in what follows, η(t) ≃ ξ(t) means C 1 ξ(t) ≤ η(t) ≤ C 2 ξ(t) for some positive constants C 1 ≤ C 2 and all t in the specified range. In this paper, to understand the effect of the mobility of the infective host on the epidemic spreading, we examine a full version of (1.7) * , where the dispersal of infective host is included. Before giving this full version, let us note that, in (1.7), since u(x, t) = 0 for x (g(t), h(t)) and , h(t)).
For i = 1, 2, suppose J i : R → R satisfy (J). Let a, b, c, d 1 , d 2 , µ 1 , µ 2 and h 0 be constants, all positive except µ 1 and µ 2 , which are assumed to be nonnegative with µ 1 + µ 2 > 0, and let the initial functions u 0 (x) and v 0 (x) satisfy (1.5). Then the full version of (1.7) can be written in the following form (1.8) We will prove the following results.
Theorem 1.2 (Spreading-Vanishing Dichotomy). Let (u, v, g, h) be the solution to (1.8) and denote h ∞ := lim t→∞ h(t) and g ∞ := lim t→∞ g(t). Then either * A full version of the local diffusion model (1.4) was recently investigated in [10], which showed that its long-time dynamics is similar to that of (1.4) though some differences occur in the criteria governing the spreading-vanishing dichotomy. In particular, when spreading happens, there exists a finite asymptotic spreading speed. Theorem 1.3 (Spreading-Vanishing Criteria). Let (u, v, g, h) be the solution of (1.8) and R 0 be given by (1.6).
(a) If R 0 ≤ 1, then vanishing always occurs. (b) If R 0 > 1, then spreading always occurs if one of the following holds: where L * > 0 is a certain critical length depending on a, b, c, d 1 , d 2 , J 1 , J 2 but independent of the initial data (u 0 , v 0 ) .
(iii) If µ 2 = 0 and all the other parameters in (1.8) are positive and fixed except d 2 , which is allowed to vary in [0, ∞), then from Lemmas 3.2 and 3.3 below it is easily seen that L * = L * (d 2 ) is strictly increasing in d 2 . Therefore part (b) of Theorem 1.3 and the result in [1] indicate that the range of parameters (a, b, c, d 1 , h 0 ) for which spreading happens regardless of the size of the initial function pare (u 0 , v 0 ), i.e., (I) or (II) above holds, is enlarged as d 2 is decreased, and such a range is maximized when d 2 = 0. Biologically, this means that reducing the mobility of the infective host increases the chance of successful spreading of the disease, which appears counter-intuitive at first look. However, such a phenomenon is not new; it arises in the local diffusion models considered in [4,10] already.
When spreading happens, the spreading profile of (1.8) can be determined by using the general results in [2]. For this purpose, we will need the following condition Theorem 1.5 (Spreading Speed). In Theorem 1.2, if spreading happens, then where c 0 > 0 is uniquely determined by the associated semi-wave problem to (1.8) (see [2, Section 1.2]).
Furthermore, in the case of accelerated spreading, we can determine the rate of accelerated spreading for a rather general class of kernel functions.
Theorem 1.6 (Rate of Accelerated Spreading). In Theorem 1.5 suppose additionally that for i = 1, 2 with µ i > 0, the kernel function J i satisfies J i (x) ≃ |x| −γ for some γ ∈ (1, 2] and |x| ≫ 1. Then for t ≫ 1, Let us note that when J i satisfies J i (x) ≃ |x| −γ for |x| ≫ 1 and for i ∈ {1, 2} such that µ i > 0, (J1) holds if and only if γ > 2. Thus Theorem 1.6 covers the exact range of γ such that accelerated spreading is possible. Note also that in condition (J1) as well as in Theorem 1.6, the condition only applies to the kernel function J i where µ i > 0. For example, if µ 2 = 0, then no extra condition on J 2 is needed apart from satisfying (J).
Problem (1.8) has an entire space version where no free boundary is involved, which has the form (1.12) Problem (1.12) has been successfully used to determine the spreading speed of the epidemic; see [11] and the references therein for many interesting results on this and related problems. For the entire space version of (1.7), see [2,12] and the references therein for more details. The local diffusion counterparts of these entire space problems have been studied more extensively; see, for example, [13][14][15]. As mentioned above, the corresponding free boundary models have the advantage of providing the exact location of the spreading front of the concerned epidemics.
The rest of the paper is organized as follows. In Section 2, we introduce the preparatory results relating to (1.8) and use them to prove Theorem 1.1. In Section 3, we gather the necessary results associated with the corresponding fixed boundary problems, which will be used to determine the longtime dynamical behavior of (1.8). In Section 4, we use the results of the previous sections to establish the vanishing-spreading dichotomy as related to the reproduction number R 0 , proving Theorems 1.2 and 1.3. Finally, Section 5 is devoted to proving the assumptions required in [2] for Theorems 1.5 and 1.6.
We would like to point out that, although (1.8) has some significantly different features from the West Nile virus model studied in [16], for example, the nature of the reaction terms in (1.8) makes any nonnegative initial function admissible while the model in [16] only allows initial functions taken from a certain bounded order interval, but many techniques of [16] can be adapted to treat (1.8), which has helped to considerably reducing the length of the current paper. Here we only provide the details of the proofs when they are very different from [16].

Existence and uniqueness, and comparison principle
In this section, we prove the well-posedness of (1.8) and some associated comparison principles. We first recall a maximum principle from [16, Lemma 3.1], which is more general than needed in this paper, but in view of possible applications elsewhere, we state it in the general form as in [16].
Let T > 0 and ξ ∈ C([0, T ]). We define the set of strict local semi-maximum points of ξ by Similarly, the set of strict local semi-minimum points of ξ is given by where J i satisfies (J). Then the following holds: ..n} and i j, then ϕ i ≥ 0 on D T for i ∈ {1, 2, ..., n}.
Proof of Theorem 1.1. Let us first consider the following slightly modified problem, with C 2 taken from Lemma 2.4: (2.8) , in view of the conditions on G, we see that they satisfy the conditions of Theorem 4.1 in [16]. Therefore (2.8) has a unique solution (u, v, g, h) defined for all t > 0.

Eigenvalue problem
For any L > 0, we consider the eigenvalue problem     The following proposition is essential for establishing the spreading and vanishing criteria in Theorem 1.3.

Fixed boundary problem
For L > 0, we define Q L = (−L, L) × (0, ∞) and consider the corresponding fixed boundary problem of (1.8): where u 0 , v 0 ∈ C([−L, L]) are nonnegative and not identically 0 simultaneously. It is well-known that fixed boundary problems such as (3.6) has a unique positive solution which is defined for all t > 0 (see, for example, Remark 4.3 in [16]). The corresponding steady state problem of (3.6) is x ∈ (−L, L).
It is called a lower solution of (3.7) if these inequalities are reversed. Proof. Due to the different nature of the reaction terms in (1.8) from the model in [16], our proof here uses rather different techniques. In particular, we will make use of the monotonicity (in time t) of the to-be-constructed lower and upper solutions and Dini's theorem.
We show next that (U, V) as well as any other positive solution of (3.7) are continuous in [−L, L]. Indeed, from the continuity of J 1 and J 2 , we easily see that From the conditions on G, we see that F(z) := (a+d 1 )(b+d 2 ) Thus from F(U(x)) = G 2 (x) + b+d 2 c G 1 (x), F ′ (z) > 0 and the fact that G 1 (x) and G 2 (x) are continuous, we obtain U(x) is continuous, which in turn implies that V(x) = a+d 1 c U(x) − G 1 (x) c is continuous. To prove uniqueness, let (Û,V) be another positive solution of (3.7). By choosing ϵ > 0 sufficiently small, we may assume that (ϵϕ 1 (x), ϵψ 1 (x)) ≤ (Û(x),V(x)) in [−L, L]. Thus the above obtained (U, V) satisfies (U, V) ≤ (Û,V).
Following the proof of [16, Proposition 3.5], we have the following result.

Spreading-vanishing dichotomy and critera
In this section, we prove Theorems 1.2 and 1.3. It is clear that h(t) and g(t) are respectively monotonically increasing and decreasing. Therefore, their limits are well-defined.

Vanishing
Here we look at cases where vanishing happens: either when the reproduction number R 0 ≤ 1 or for sufficiently small initial data (u 0 , v 0 ).

4)
and hence vanishing happens, where m 0 := min{d 1 , d 2 c b }. Proof. Since J 1 (x − y)u(x, t)dydx ≤ 0, and the same holds when (u, J 1 ) is replaced by (v, J 2 ). By the above, we obtain that d dt Integrating the above from 0 to t gives us (4.4). Then by Lemma 4.1, we obtain the vanishing result. □ Now for initial data (u 0 , v 0 ) small enough, we show that vanishing also occurs.
is sufficiently small, then vanishing happens.
We note that the following lemma implies that if h ∞ − g ∞ = ∞ holds, then we must have that h ∞ = −g ∞ = +∞. Proof. The proof of this lemma is similar to the proof of Lemma 4.10 in [16]. Since the modifications are obvious, we omit the details. □

Spreading
In this section, we look at cases where spreading happens.
Proof of Lemma 4.6. We see that for some t ≥ t 0 , by Corollary 3.5, we have λ 1 (g(t), h(t)) < 0 for t ≥ t 0 . Then we derive the contradiction as in Lemma 4.1. By Lemma 4.5, we further obtain that −g ∞ = h ∞ = ∞. Then by Lemma 4.2, we must have R 0 > 1, which guarantees the existence of the positive equilibrium (u * , v * ). It remains to show (4.6). Let us first consider the limit superior of the solution. Let (u, v) be the unique positive solution of the following ODE problem: Since R 0 > 1, we have lim t→∞ (u(t), v(t)) = (u * , v * ). We then note that and u(0) ≥ u 0 (x), v(0) ≥ v 0 (x); so by the comparison principle in Lemma 2.3, we have (u(x, t), v(x, t)) ≤ (u(t), v(t)) for g(t) < x < h(t) and t > 0.
Proof. Similar to the calculations in the proof of Lemma 4.2, by setting m 0 := max{d 1 , d 2 c b }, we obtain for t > 0, Suppose that h ∞ − g ∞ < ∞; then in view of Lemma 4.1 and Remark 4.7, by letting t → ∞ in the above inequality, we get However, this is patently false in the case µ > µ 0 := 2m 0 (L * − h 0 ) This completes the proof. □

Spreading speed
In this section, we consider the asymptotic spreading speed when spreading happens in our system (1.8). As such, we necessarily have that R 0 > 1.
is linear for u < u * close to u * and v < v * close to v * .
(f 4 ): The solution of the corresponding problem (1.12) with initial function pair (u 0 , v 0 ) nonnegative, bounded and not identically (0, 0) is positive and globally defined, and as time t → ∞, it converges to (u * , v * ) locally uniformly for x ∈ R.